

A003641


Number of genera of imaginary quadratic field with discriminant k, k = A039957(n).
(Formerly M0061)


3



1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 2, 1, 1, 4, 2, 1, 2, 1, 4, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 4, 2, 2, 2, 2, 1, 2, 1, 4, 1, 1, 2, 2, 4, 1, 1, 2, 1, 4, 1, 1, 1, 1, 2
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OFFSET

1,4


COMMENTS

In other words, this is the number of genera of those imaginary quadratic fields that have a discriminant which is odd and fundamental. The discriminant will be squarefree and of the form 4n+1.  Andrew Howroyd, Jul 25 2018


REFERENCES

D. A. Buell, Binary Quadratic Forms. SpringerVerlag, NY, 1989, pp. 224241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=1..105.
Index entries for sequences related to quadratic fields


FORMULA

a(n) = 2^(omega(A039957(n))  1).  Jianing Song, Jul 24 2018


EXAMPLE

a(4) = 2 because 15 = A039957(4) and the number of genera of the quadratic field with discriminant 15 is 2.  Andrew Howroyd, Jul 25 2018


MATHEMATICA

2^(PrimeNu[Select[Range[1000], Mod[#, 4] == 3 && SquareFreeQ[#]&]]  1) (* JeanFrançois Alcover, Jul 25 2019, after Andrew Howroyd *)


PROG

(PARI) for(n=1, 1000, if(n%4==3 && issquarefree(n), print1(2^(omega(n)  1), ", "))) \\ Andrew Howroyd, Jul 24 2018


CROSSREFS

Cf. A001221 (omega), A003640, A003642, A039957.
Sequence in context: A317934 A279848 A001826 * A165190 A025890 A316975
Adjacent sequences: A003638 A003639 A003640 * A003642 A003643 A003644


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Mira Bernstein


EXTENSIONS

Name clarified by Jianing Song, Jul 24 2018


STATUS

approved



